Definable Combinatorics of Graphs and Equivalence Relations

• Main Author: Meehan, Connor George Walmsley
• Format: Others
• Published: 2018
• Online Access: https://thesis.library.caltech.edu/11006/1/meehan_connor_2018.pdf; Meehan, Connor George Walmsley (2018) Definable Combinatorics of Graphs and Equivalence Relations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/45E4-MC27. https://resolver.caltech.edu/CaltechTHESIS:06012018-160828760 <https://resolver.caltech.edu/CaltechTHESIS:06012018-160828760>

<p>Let <b>D</b> = (<i>X</i>, <i>D</i>) be a Borel directed graph on a standard Borel space <i>X</i> and let <i>χ_B</i>(<b>D</b>) be its Borel chromatic number. If <i>F</i>₀, …, <i>F</i>_<i>n</i>-1: <i>X</i> → <i>X</i> are Borel functions, let <b>D</b>_{<i>F</i>₀, …, <i>F</i>_<i>n</i>-1} be the directed graph that they generate. It is an open problem if <i>χ_B</i>(<b>D</b>_{<i>F</i>₀, …, <i>F</i>_<i>n</i>-1}) ∈ {1, …, 2<i>n</i> + 1, ℵ₀}. Palamourdas verified the foregoing for commuting functions with no fixed points. We show here that for commuting functions with the property that there is a path from each <i>x</i> ∈ <i>X</i> to a fixed point of some <i>F_j</i>, there exists an increasing filtration <i>X</i> = ⋃_{<i>m</i> &lt; <i>ω</i>} <i>X_m</i> such that <i>χ_B</i>(<b>D</b>_{<i>F</i>₀, …, <i>F</i>_<i>n</i>-1}↾ <i>X</i>_m) ≤ 2<i>n</i> for each <i>m</i>. We also prove that if <i>n</i> = 2 in the previous case, then <i>χ_B</i>(<b>D</b>_{<i>F</i>₀, <i>F</i>₁}) ≤ 4. It follows that the approximate measure chromatic number <i>χ^ap_M</i>(<b>D</b>) ≤ 2<i>n</i> + 1 when the functions commute.</p> <p>If <i>X</i> is a set, <i>E</i> is an equivalence relation on <i>X</i>, and <i>n</i> ∈ <i>ω</i>, then define [<i>X</i>]^<i>n</i>_<i>E</i> = {(<i>x</i>₀, ..., <i>x</i>_{<i>n</i> - 1}) ∈ ^<i>n</i><i>X</i>: (∀<i>i</i>,<i>j</i>)(<i>i</i> ≠ <i>j</i> → ¬(<i>x_i</i> <i>E</i> <i>x_j</i>))}. For <i>n</i> ∈ <i>ω</i>, a set <i>X</i> has the <i>n</i>-Jónsson property if and only if for every function <i>f</i>: [<i>X</i>]^<i>n</i>₌ → <i>X</i>, there exists some <i>Y</i> ⊆ <i>X</i> with <i>X</i> and <i>Y</i> in bijection so that <i>f</i>[[<i>Y</i>]^<i>n</i>₌] ≠ <i>X</i>. A set <i>X</i> has the Jónsson property if and only for every function <i>f</i> : (⋃_{<i>n</i> ∈ <i>ω</i>} [<i>X</i>]^<i>n</i>₌) → <i>X</i>, there exists some <i>Y</i> ⊆ <i>X</i> with <i>X</i> and <i>Y</i> in bijection so that <i>f</i>[⋃_{<i>n</i> ∈ <i>ω</i>} [<i>Y</i>]^<i>n</i>₌] ≠ <i>X</i>. Let <i>n</i> ∈ <i>ω</i>, <i>X</i> be a Polish space, and <i>E</i> be an equivalence relation on <i>X</i>. <i>E</i> has the <i>n</i>-Mycielski property if and only if for all comeager <i>C</i> ⊆ ^<i>n</i><i>X</i>, there is some Borel <i>A</i> ⊆ <i>X</i> so that <i>E</i> ≤_<i>B</i> <i>E</i> ↾ <i>A</i> and [<i>A</i>]^<i>n</i>_<i>E</i> ⊆ <i>C</i>. The following equivalence relations will be considered: <i>E</i>₀ is defined on ^<i>ω</i>2 by <i>x</i> <i>E</i>₀ <i>y</i> if and only if (∃<i>n</i>)(∀<i>k</i> &gt; <i>n</i>)(<i>x</i>(<i>k</i>) = <i>y</i>(<i>k</i>)). <i>E</i>₁ is defined on ^<i>ω</i>(^<i>ω</i>2) by <i>x</i> <i>E</i>₁ <i>y</i> if and only if (∃<i>n</i>)(∀<i>k</i> &gt; <i>n</i>)(<i>x</i>(<i>k</i>) = <i>y</i>(<i>k</i>)). <i>E</i>₂ is defined on ^<i>ω</i>2 by <i>x</i> <i>E</i>₂ <i>y</i> if and only if ∑{¹⁄_{(<i>n</i> + 1)}: <i>x</i>(<i>n</i>) ≠ <i>y</i>(<i>n</i>)} &lt; ∞. <i>E</i>₃ is defined on ^<i>ω</i>(^<i>ω</i>2) by <i>x</i> <i>E</i>₃ <i>y</i> if and only if (∀<i>n</i>)(<i>x</i>(<i>n</i>) <i>E</i>₀ <i>y</i>(<i>n</i>)). Holshouser and Jackson have shown that ℝ is Jónsson under AD. The present research will show that <i>E</i>₀ does not have the 3-Mycielski property and that <i>E</i>₁, <i>E</i>₂, and <i>E</i>₃ do not have the 2-Mycielski property. Under ZF + AD, ^<i>ω</i>2/<i>E</i>₀ does not have the 3-Jónsson property.</p> <p>Let <b>G</b> = (<i>X</i>, <i>G</i>) be a graph and define for <i>b</i> ≥ 1 its <i>b</i>-fold chromatic number <i>χ</i>^(<i>b</i>)(<b>G</b>) as the minimum size of <i>Y</i> such that there is a function <i>c</i> from <i>X</i> into <i>b</i>-sets of <i>Y</i> with <i>c</i>(<i>x</i>) ∩ <i>c</i>(<i>y</i>) = ∅ if <i>x</i> <i>G</i> <i>y</i>. Then its fractional chromatic number is <i>χ</i>^<i>f</i>(<b>G</b>) = inf_<i>b</i> ^{<i>χ</i>^(<i>b</i>)(<b>G</b>)}⁄_<i>b</i> if the quotients are finite. If <i>X</i> is Polish and <b>G</b> is a Borel graph, we can also define its fractional Borel chromatic number <i>χ</i>^<i>f</i>_<i>B</i>(<b>G</b>) by restricting to only Borel functions. We similarly define this for Baire measurable and <i>μ</i>-measurable functions for a Borel measure <i>μ</i>. We show that for each countable graph <b>G</b>, one may construct an acyclic Borel graph <b>G</b>' on a Polish space such that <i>χ</i>^<i>f</i>_<i>BM</i>(<b>G</b>') = <i>χ</i>^<i>f</i>(<b>G</b>) and <i>χ</i>_<i>BM</i>(<b>G</b>') = <i>χ</i>(<b>G</b>), and similarly for <i>χ</i>^<i>f</i>_<i>μ</i> and <i>χ</i>_<i>μ</i>. We also prove that the implication <i>χ</i>^<i>f</i>(<b>G</b>) = 2 ⇒ <i>χ</i>(<b>G</b>) = 2 is false in the Borel setting.</p>